Magnetic Moment Estimation Algorithm Based on Convolutional Neural Network

Abstract

Estimating the magnetic moment of a magnetic target is a key aspect of magnetic anomaly detection when accurately locating the target. Current methods primarily rely on vector magnetic field signals or gradient signals, which face challenges such as steering errors and the high costs associated with sensor arrays. This paper proposes a magnetic moment estimation algorithm that combines a scalar magnetic field sensor and the three components of the local geomagnetic field with a convolutional neural network (CNN). The simulation results demonstrate that the proposed algorithm performs well in noise environments with a signal-to-noise ratio (SNR) greater than −5 dB. At an SNR of −5 dB, the relative error in the magnetic moment magnitude is 4.92%, while the absolute errors in the magnetic moment declination and inclination are $4.73°$ and $1.54°$, respectively. The vector angle difference is $2.7°$. The algorithm achieves high estimation accuracy even with a scalar sensor and under poor noise conditions, offering a novel and effective approach for future magnetic moment estimation research.

1. Introduction

Magnetic anomaly detection (MAD) technology is used to detect and localize local magnetic targets by analyzing magnetic anomaly signals [1]. It can be categorized into aeromagnetic, ground-based, and satellite-based surveys, depending on the platform employed for detection. Among these, aeromagnetic surveys involve placing magnetic detectors on aircraft to detect magnetic anomalies. This method offers advantages such as a wide detection range, high operational efficiency, and precise detection accuracy [2]. Aeromagnetic surveys are extensively utilized in various fields, including mineral exploration [3], where they are used for geological mapping as well as anomaly detection, anti-submarine early warning [4,5], and unexploded ordnance detection [6,7], among others.

In airborne MAD surveys, estimating the magnetic moment of magnetic targets from detection data is a critical step [8,9], as it can significantly reduce the false alarm rate [10]. Wynn inverted the magnetic dipole moment information using the magnetic field vector and magnetic gradient signals measured by a magnetic gradiometer and sensor array, thereby estimating the magnetic moment of the target [11]. Wiegert employed a multi-tensor gradiometer and fluxgate array for the triangulation of the magnetic target and proposed the STAR algorithm to estimate the magnetic moment [12]. However, the STAR algorithm is affected by issues such as significant measurement noise and inherent errors due to the tensor contraction assumption. To address these limitations, Liu enhanced the direction vector selection mechanism relying on rotation invariance and optimized the distance estimation, thereby improving positioning accuracy [13]. Sheinker introduced a nonlinear optimization approach using simulated annealing relying on the magnetic dipole model [14]. Ege examined the geometric characteristics of magnetic targets by studying how magnetic materials influence the vertical component of the geomagnetic field [15]. Qin et al. developed an algorithm grounded in a full magnetic gradient orthogonal basis function, which can accurately estimate the vector magnetic moment of the target under low SNR conditions [16]. Ou, Qiu, and colleagues integrated the target speed, movement direction, the distance between the sensor and the magnetic target, and the magnetic dipole model to construct a magnetic dipole motion model, which they used to identify the magnetic target by comparing the waveforms of the model signal with those of the collected signal [17].

Currently, most magnetic moment estimation algorithms rely on magnetic field data collected by vector magnetic sensors and magnetic gradiometers. However, vector magnetic sensors are prone to issues such as large steering errors [18] and low sensitivity, making them less suitable for mobile platforms. In contrast, scalar magnetic sensors offer higher sensitivity and stability [19], making them ideal for use on high-speed airborne platforms. Fan utilized a scalar magnetic sensor array combined with a particle swarm optimization algorithm to perform target detection [20]. Du proposed an estimation method relying on magnetic field data from multiple scalar magnetic sensors, employing orthogonal basis decomposition and average filtering techniques [21]. While these algorithms offer valuable insights, they require a larger number of sensors and more complex data processing, placing high demands on the platform. Jin introduced a correlation detection method established on normalized magnetic moments. This approach performs well for estimating the magnetic moment modulus under low SNR conditions, but it struggles with accurately estimating the direction of the magnetic moment [22]. This paper proposes a magnetic moment estimation algorithm that integrates a scalar magnetic sensor with CNN. The input of algorithm includes the three components of 2D geomagnetic field and 2D scalar magnetic anomaly signal, while the output consists of magnetic moment vector. This approach ensures accurate magnetic moment estimation while maintaining low-cost applicability.

2. Materials and Methods

2.1. Magnetic Anomaly Detection

2.1.1. Principles and Methods

Magnetic sensors can be classified into scalar magnetic sensors and vector magnetic sensors based on the type of field quantity they measure. A vector magnetic field sensor measures the three components of the total magnetic field, providing directional information [23]. However, vector magnetic field sensors are susceptible to steering errors, making them less suitable for use on moving vehicles. In contrast, scalar magnetic field sensors, which measure the modulus of the total magnetic field, are widely used on flying platforms due to their superior stability.

As shown in Figure 1, let ${\mathbf{H}}_T$, ${\mathbf{H}}_E$, and ${\mathbf{H}}_r$ represent the total magnetic field, the geomagnetic field, and the magnetic anomaly signal, and the two angles $\theta$ and $\phi$ represent the angles between different vectors, respectively [24]. Using the law of cosines, we can derive the following equation:

$$\begin{array}{c}\hfill {\mathbf{H}}_T={\mathbf{H}}_E+{\mathbf{H}}_r\end{array}$$

(1)

$$\begin{array}{cc}\hfill |{\mathbf{H}}_T|& =\sqrt{|{\mathbf{H}}_E{|}^2+|{\mathbf{H}}_r{|}^2-2|{\mathbf{H}}_E\left|\right|{\mathbf{H}}_r|cos(\pi -\phi )}\hfill \\ \hfill & =\left|{\mathbf{H}}_E\right|\sqrt{1+{\displaystyle \frac{|{\mathbf{H}}_r{|}^2+2|{\mathbf{H}}_E\left|\right|{\mathbf{H}}_r|cos\left(\phi \right)}{|{\mathbf{H}}_E{|}^2}}}.\hfill \end{array}$$

(2)

In aeromagnetic surveys, since the magnitude of the geomagnetic field is significantly higher than that of the magnetic anomaly signal, the last term in Equation (2) can be represented as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{|{\mathbf{H}}_r{|}^2+2|{\mathbf{H}}_E\left|\right|{\mathbf{H}}_r|cos\left(\phi \right)}{|{\mathbf{H}}_E{|}^2}}& \to 0\hfill \end{array}$$

(3)

In Equation (3), the total magnetic field signal can be represented as follows:

$$\begin{array}{cc}\hfill \underset{x\to 0}{lim}{(1+x)}^{\alpha }& \sim 1+\alpha ·x\hfill \end{array}$$

(4)

from which

$$\begin{array}{c}|{\mathbf{H}}_T|\approx \left|{\mathbf{H}}_E\right|+|{\mathbf{H}}_r|cos\phi +{\displaystyle \frac{|{\mathbf{H}}_r{|}^2}{2|{\mathbf{H}}_E|}}.\end{array}$$

(5)

As mentioned earlier, the last term in Equation (5) can be omitted. Therefore, the measured magnetic field signal can be expressed as follows:

$$\begin{array}{c}|{\mathbf{H}}_T|\approx |{\mathbf{H}}_E|+|{\mathbf{H}}_r|cos\phi .\end{array}$$

(6)

According to Equation (6), the total magnetic field signal can be expressed as the magnitude of the geomagnetic field along with the projection of the magnetic anomaly signal in the direction of the geomagnetic field. The process of generating the magnetic anomaly signal and the principle of aeromagnetic measurement are illustrated in Figure 2.

2.1.2. Geomagnetic Field, Magnetic Moment, and Magnetic Anomaly Signals

As discussed earlier, the magnetic anomaly field is determined by both the geomagnetic field vector and the target magnetic field. To offer a clearer explanation of the magnetic moment angular characteristics, we will now describe the associated angular parameters. According to Figure 3, the angle formed between the magnetic moment and its projection onto the XY plane is defined as the inclination angle, denoted as $I_M$. Meanwhile, the angle between the projection of the magnetic moment onto the XY plane and the X-axis is defined as the declination angle, denoted as $D_M$.

Assume that the direction of the geomagnetic field is parallel to the Y-axis, the target is located at $(0,0,0)$, and the observation range spans from $-30$ to 30 along both axes, with a sampling point interval of $0.1$. The observation plane is set at $z=10$, and the directions of the total magnetic moment vector are $(1,0,0)$, $(0,1,0)$, $(0,0,1)$, and $(1,1,1)$. The magnetic field intensity is provided in nanoteslas (nT).

We simulated the magnetic anomaly field under varying magnetic moment orientations within the coordinate system, allowing us to examine the relationship between the magnetic moment direction, the geomagnetic field direction, and the resulting magnetic anomaly field.

The simulation results are shown in Figure 4, which illustrates the impact of changes in the magnetic moment direction on the magnetic anomaly signal while keeping the magnetic moment magnitude constant. Examples are highlighted in the figure, with the red dashed lines representing the magnetic anomaly signal. When the total magnetic moment is orthogonal to the geomagnetic field and parallel to the observation surface, the magnetic anomaly vector exhibits symmetry along the target center, with peaks and troughs symmetrically distributed around the center. In contrast, when the magnetic moment is parallel to the geomagnetic field in the same direction, the magnetic anomaly vector at the center of the observation surface aligns with the geomagnetic field but points in the opposite direction. This results in the largest negative projection of the magnetic anomaly field along the geomagnetic direction, and the angle between the magnetic anomaly vectors on either side of the center and the positive direction of the geomagnetic field is less than $90°$, with symmetry around the center. As a result, the total magnetic anomaly field forms a single trough distribution.

When the magnetic moment is orthogonal to the geomagnetic field and perpendicular to the observation surface, the magnetic anomaly vector at the center is also perpendicular to the geomagnetic field, resulting in a zero projection along the geomagnetic field direction. This configuration generates a magnetic anomaly field with peaks and troughs symmetrically distributed around the target origin. Lastly, when the magnetic moment is angled relative to the geomagnetic field, the total magnetic anomaly field exhibits a double peak—one positive and one negative—where the peak line is offset from the target origin.

These simulation results confirm that the vector relationship between the magnetic anomaly field and the geomagnetic field primarily governs the distribution of the total magnetic anomaly field signal.

2.2. Magnetic Dipole Model

In magnetic anomaly detection research, the size of the magnetic anomaly target is typically much smaller than the distance between the measurement point and the target. When this distance is 2 to 3 times greater than the target size, the magnetic material can be approximated as a magnetic dipole [25]. Under this assumption, the magnetic field of the dipole can serve as an equivalent representation of the target magnetic field.

Relying on the condition that the magnetic target can be equivalently represented as a magnetic dipole, the magnetic anomaly signal is derived using the magnetic dipole model. A coordinate system is established according to Figure 5, with the magnetic target placed at the origin. The coordinates of the magnetic field sensor are denoted as ${(x,y,z)}^T$, and the distance between the magnetic field sensor and the magnetic target is r. The magnetic target is equivalently represented as a magnetic dipole with a magnetic moment $\mathbf{M}={(m_x,m_y,m_z)}^T$. The expression for the magnetic anomaly signal is given as follows:

$$\begin{array}{c}\hfill S={\displaystyle \frac{{\mu }_0}{4\pi r^5}}\left[\left(3x^2+3xy+3xz-r^2\right)m_x+\left(3xy+3y^2+3yz-r^2\right)m_y+\left(3xz+3yz+3z^2-r^2\right)m_z\right].\end{array}$$

(7)

By substituting Equation (7) into Equation (6), the magnetic anomaly signal S detected by the scalar magnetic sensor can be expressed as follows:

$$S=|{\mathbf{H}}_T|-|{\mathbf{H}}_E|=|{\mathbf{H}}_r|cos\alpha$$

(8)

which expands to:

$$\begin{array}{cc}\hfill S={\displaystyle \frac{{\mu }_0}{4\pi |{\mathbf{H}}_E|r^5}}\{& [(3x^2-r^2)H_{E_x}+3xy{H}_{E_y}+3xz{H}_{E_z}]m_x\hfill \\ \hfill & +\left[3xy{H}_{E_x}+(3y^2-r^2)H_{E_y}+3yz{H}_{E_z}\right]m_y\hfill \\ \hfill & +\left[3xz{H}_{E_x}+3yz{H}_{E_y}+(3z^2-r^2)H_{E_z}\right]m_z\}.\hfill \end{array}$$

(9)

As seen from the above equation, there are unknown quantities, such as the target position, which prevent the direct calculation of the target magnetic moment by inverting the magnetic anomaly signal. A direct, closed-form linear relationship between the target magnetic moment and the magnetic anomaly signal cannot be established. To address this challenge, a nonlinear model can be used to solve the problem.

Neural networks have undoubtedly become a crucial tool in this context, as they excel at modeling relationships between data that are difficult to describe with linear models [26]. Therefore, we chose to use a 2D CNN to model the relationship between the target magnetic anomaly signal, the geomagnetic field, and the target magnetic moment. Previous studies have employed CNN to process target data and localize the target [27]. This paper proposes a fitting network based on a 2D CNN to establish the relationship between the target magnetic moment and the magnetic anomaly signal.

3. Model and Experiment

3.1. Neural Network Model

CNN comprise deep learning models particularly well-suited for processing data with grid-like structures, such as images, videos, audio, and time series data [28]. In this paper, a 2D CNN architecture is introduced to estimate the target magnetic moment, relying on the three-component geomagnetic field signal and the scalar magnetic anomaly signal obtained from the measured target. Compared to fully connected neural networks, the CNN is better at capturing local information features. By convolving the target magnetic anomaly signal with the geomagnetic field, spatial local features within the signal are extracted. Furthermore, due to the use of convolutional kernels, memory consumption and computational costs are reduced.

The structure of the CNN model proposed in this paper is illustrated in Figure 6 composed of three convolutional layers, two pooling layers, and one fully connected layer, with the detailed settings provided in

Table 1. Detailed settings of the neural network.

Layer	Parameter Settings
Conv1	Conv2D: Kernel number: 16; kernel size: $3\times 3$; strides: 1; padding: “same”; ReLU
Conv2	Conv2D: Kernel number: 32; kernel size: $3\times 3$; strides: 1; padding: “same”; ReLU
Conv3	Conv2D: Kernel number: 64; kernel size: $3\times 3$; strides: 1; padding: “same”; ReLU
Pool1	MaxPooling2D: pool size: $2\times 2$; strides: 2; padding: “same”
Pool2	MaxPooling2D: pool size: $2\times 2$; strides: 2; padding: “same”
Flatten	Flatten the input to 1D vector; ReLU

. The $3\times 3$ convolution kernel, compared to larger kernels like $5\times 5$, incurs lower computational cost while achieving a larger receptive field through the stacking of multiple $3\times 3$ kernels. The design of the three convolution layers aims to progressively extract features from low order to high order, striking a balance between feature extraction and computational efficiency. As the number of convolutional layers increases, the receptive field expands, enhancing the network’s ability to capture local features [29]. The network also includes two pooling layers, which downsample the convolutional data to reduce computational complexity and improve the model’s generalization ability. The network achieves a lightweight design while maintaining strong predictive performance.

The model described in this article was trained on a laptop computer. The training time was approximately 14 min, with 50 training epochs, a learning rate of ${10}^{-3}$, and a batch size of 5. The specifications of the computer used for training are as follows:

Processor: 13th Gen Intel(R) Core(TM) i5-13500HX 2.50 GHz;

GPU: NVIDIA GeForce RTX 4060 Laptop GPU 8 GB;

Memory: 16 GB.

3.2. Dataset

The 2D CNN proposed in this paper uses the three components of the geomagnetic field and the scalar magnetic anomaly signal as input samples. The network consists of three convolutional layers designed to extract feature information from the input samples, with the labels corresponding to the components of the target magnetic moment.

A dataset is generated in MATLAB using the magnetic dipole model. The input dataset consists of the three components of the geomagnetic field from the model and the scalar magnetic anomaly signal. The label data include the three components of the magnetic moment. These three components are defined within the same coordinate system, specifically using the North–East–Down (NED) coordinate system, to facilitate subsequent work. Given the small horizontal detection distance, the plane gradient of the geomagnetic field can be neglected [30], and it is assumed that the geomagnetic field signal remains constant within the detection range. The dataset is generated according to the parameters specified in

Table 2. Simulation target parameters.

Parameter	Settings
Local geomagnetic field setting (Modulus, Inclination, Deflection)	$(5\times {10}^4\mathrm{nT},45°,-5°)$
Target magnetic moment setting (Modulus, Inclination, Deflection)	(${10}^4\sim {10}^5\mathrm{A}·{\mathrm{m}}^2,-90°\sim 90°,$$-180°\sim 180°$) (Step Length: ${10}^4\mathrm{A}·{\mathrm{m}}^2,30°,45°$)
Local size	$4\times 4\mathrm{km}$
Spatial sampling rate	$0.1$ sampling point/m
Relative height	$500\sim 1000\mathrm{m}$ (Step Length: $100\mathrm{m}$)

:

1.

The magnitude of the geomagnetic field is 50,000 nT, with a geomagnetic field inclination of $45°$ and a declination of $-5°$;

2.

The simulation range is $4\mathrm{km}\times 4\mathrm{km}$, with a spatial sampling rate of 0.1 sampling points per meter;

3.

The height varies from 500 m to 1000 m in 100 m steps;

4.

The target magnetic moment modulus ranges from ${10}^4$ to ${10}^5$ A·m², with a step size of ${10}^4$ A·m²;

5.

The target magnetic moment declination ranges from $-180°$ to $180°$, with a step size of $45°$. The magnetic moment inclination ranges from $-90°$ to $90°$, with a step size of $30°$.

The dataset used in this paper consists of 1800 samples, each represented by a $401\times 401\times 4$ matrix, with the output label being a three-dimensional vector. The dataset is divided into a training set (70%), a validation set (15%), and a test set (15%). The samples include scalar magnetic anomaly signals and three-component geomagnetic field signals, generated from various heights, target magnetic moment magnitudes, and magnetic moment angle configurations.

The independent test set ensures that the model is not overfitted to the training data, offering a more accurate evaluation of its final performance and generalization ability. This approach highlights the model’s capacity to handle new data in real-world scenarios.

During aeromagnetic operations, magnetic anomaly signals detected by the magnetic sensor are subject to noise. The performance of the model is evaluated under different noise levels by applying varying SNRs to the dataset. The noise primarily results from the aeromagnetic environment, with the flight vehicle being the main factor influencing the noise level [31]. As a result, the noise impacts both the training and test set data.

Gaussian white noise with SNRs ranging from $-30$ dB to 50 dB were added to the dataset. Figure 7 illustrates this, where Figure 7a represents the dataset without noise, and Figure 7b shows the dataset after noise has been added.

Currently, the noise floor of scalar magnetic field sensors is quite low. For instance, the Quspin QZFM Gen-3 scalar magnetic field sensor has a noise floor of $15\mathrm{fT}/\sqrt{\mathrm{Hz}}$, which has little impact on magnetic field data collection. As a result, the noise from scalar magnetic field sensors does not significantly affect the data, making it acceptable. Introducing noise simulates the impact of environmental noise on the magnetic field data. The environmental magnetic field noise would influence target magnetic moment estimation by inferencing the network’s feature extraction.

3.3. Performance Indicators

This paper uses the Mean Absolute Percentage Error (MAPE) to evaluate the magnetic moment modulus. MAPE represents the percentage of the error relative to the true value, is dimensionless, and provides an intuitive measure of the prediction error in relation to the actual value. The calculation equation for MAPE is given in Equation (10), where represents the true value of the modulus and represents the predicted value of the magnetic moment.

$$E_M={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|{\displaystyle \frac{M_i-{\widehat{M}}_i}{M_i}}\right|\times 100\%$$

(10)

The Mean Absolute Error (MAE) is a simple and intuitive metric that directly reflects the average difference between the predicted and true values. It effectively measures the performance of the algorithm, as it provides the same impact for the same error magnitude, regardless of the actual value [32]. Therefore, MAE is used to measure the errors of declination and inclination. The calculation of the prediction error for the deflection and inclination of the magnetic moment is represented by Equation (11).

As in previous sections, $D_i$ and $I_i$ represent the true values of the magnetic moment deflection and inclination, while ${\widehat{D}}_i$ and ${\widehat{I}}_i$ represent the predicted values of these parameters.

$$E_D={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|D_i-{\widehat{D}}_i\right|,E_I={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|I_i-{\widehat{I}}_i\right|$$

(11)

4. Results

4.1. Ideal State

Seventy percent of the dataset is used for training the network. It is important to note that the selection of the training set is random. The following are the test results after the model training is completed. The test results show that the error in the magnetic moment modulus is 1.85%, and the average absolute errors for the magnetic moment deflection and inclination are $1.83°$ and $0.75°$, respectively. The average absolute error between the predicted magnetic moment and the test ranging angle is approximately $1.12°$, the results are shown in

Table 3. Fitting error.

Performance Metrics	Error
MAPE of magnetic moment magnitude	$1.85\%$
MAE of magnetic moment deflection	$1.83°$
MAE of magnetic moment inclination	$0.75°$
MAE of the angular difference between two vectors	$1.12°$

. The model is demonstrated to possess a strong magnetic moment fitting capability, effectively capturing the underlying relationship between the input data and the target magnetic moment. This capability allows the model to achieve highly accurate predictions on the test set, maintaining a relatively small error margin.

When the inclination angle of the test magnetic moment is close to 90°, small changes in the X and Y components of the magnetic moment can cause significant variations in the magnetic moment deflection, leading to drastic fluctuations in the deflection error. In contrast, the magnetic moment inclination changes less drastically during the calculation process, so the magnetic moment inclination error is smaller than the magnetic moment deflection error.

Since the data are randomly selected, the test set data are arranged for clarity, as shown in Figure 8, Figure 9 and Figure 10.

4.2. Adding Noise

During aeromagnetic operations, noise inevitably affects the accuracy of the measurements. The performance of the model is evaluated under different noise levels by applying varying signal-to-noise ratios (SNRs) to the dataset. The noise primarily originates from the aeromagnetic environment, with the sensor carrier playing a crucial role as the dominant factor influencing the noise level. This influence is particularly evident on moving platforms, where vibrations, orientation changes, and equipment onboard generate additional noise. As a result, noise affects both the training data and the test set.

Gaussian white noise with an SNR ranging from $-30$ to 50 dB is added to the dataset. The model is then trained using the data segmentation and training methods described above. The following results are obtained: the test results of the model trained with samples at an SNR of $-5$ dB are shown in Figure 11, Figure 12 and Figure 13.

From Figure 14 and Figure 15, it can be observed that as the SNR increases, the error generally decreases and eventually stabilizes. When the SNR = $-5$ dB, the errors in the three components of the magnetic moment are approximately 7%. The relative error in the magnetic moment magnitude is 4.92%, while the absolute errors in the magnetic moment declination and inclination are $4.73°$ and $1.54°$, respectively. The vector angle difference is $2.7°$.

As the model is trained with different samples, the prediction error gradually stabilizes, demonstrating the ability of the model to perform effectively in harsh noise environments. These samples are augmented with noise of gradually increasing SNR, further validating robustness and adaptability of the model to varying levels of noise interference. This behavior highlights potential of the model for reliable performance in real-world scenarios where noise conditions are unpredictable and challenging.

5. Discussion

When calculating the magnetic moment magnitude from the three components, the error in the magnitude may be smaller than the errors in the individual components if the error signs are opposite. Simulation experiments indicate that when the SNR is greater than −5 dB, the model’s performance remains relatively stable. The results indicate that the model proposed upon maintains a reasonable level of accuracy even under low SNR conditions. An advantage of this algorithm is that it relies entirely on scalar magnetic field data, enabling the aircraft to operate with just a single scalar magnetic field sensor, thus reducing hardware requirements. Another advantage is that the model allows for the direct estimation of the target’s magnetic moment size and direction. Additionally, the model demonstrates strong stability in the presence of environmental noise, maintaining high accuracy in magnetic moment estimation even under harsh conditions.

Compared to the STAR algorithm, which requires magnetic field gradient signals from multiple directions around the target, this model uses a simpler signal format and requires fewer sensors. While traditional linear algorithms may struggle to capture the complex relationship between the target magnetic moment and the magnetic anomaly signal, CNN is capable of learning this intricate mapping through activation functions. Furthermore, enhanced stability is provided by the CNN, and it is less sensitive to noise.

However, due to the two-dimensional structure of the input data, the application scenario may be somewhat limited, with the model primarily suited for the magnetic moment estimation of static targets. In practical applications, multiple factors must be considered to ensure the algorithm’s effectiveness. For example, when the target size or number is unclear, or when the sensor is positioned too close to the target, the target may no longer be accurately represented by a magnetic dipole model, leading to inaccurate magnetic moment estimation. Furthermore, the scanning area should be designed to capture the most complete signal possible. It is anticipated that future improvements will help reduce the limitations caused by the data structure.

6. Conclusions

This paper presents a magnetic moment fitting model based on a CNN and scalar magnetic anomaly signals, aimed at uncovering the complex relationship between magnetic anomaly signals, geomagnetic field signals, and the target magnetic moment. The dataset used for training the model includes 2D magnetic anomaly signals and three-component geomagnetic field signals, generated under various heights and magnetic moment conditions, with the three components of the magnetic moment serving as network labels.

Additionally, Gaussian white noise with varying SNRs was added to the dataset. After training the model, the error demonstrated an overall decrease and stabilized as the SNR improved. When the SNR = $-5$ dB, the model achieved the relative error in the magnetic moment magnitude is 4.92%, while the absolute errors in the magnetic moment declination and inclination are $4.73°$ and $1.54°$, respectively. The vector angle difference is $2.7°$. These results highlight the model stability, even in challenging environments.

The primary goal of this model is to minimize the error in magnetic moment fitting under conditions with a scalar magnetic sensor and low SNR, effectively addressing the instability of vector magnetic sensors caused by steering discrepancies. By doing so, it enhances the efficiency of magnetic moment fitting across diverse environmental conditions. The model offers valuable insights into the precise fitting of magnetic moments under low SNR conditions, providing practical strategies for researchers in applications such as unexploded ordnance detection [33], urban pipeline installation [34], and medical robotics [35].